I did a search on the net and found the term “base 10” all over the place. What does it mean?

An apparently annoying question:
“Does the 1 in 10 stand for the number 10’s in 10?”.

The interpretation of 10 in the system described as “Base 10” depends on the base of the system, so what is it? How do I find out?

We have here a logical problem. The term “Base 10” as a definition is self referential. It is more subtle than this definition of a straight line:

“A straight line is a line which is straight”.

The problem arises from the almost universal confusion between the two things:
1: The name of a number, in this case “ten” is supposedly implied
2: The symbols representing a number, in this case 10 in the base ten system”

So the answers to the questions “What is it? How do I find out?” above are “Unknown” and “You can’t”

Writing “Base 10” when you mean “Base ten” is probably the first step in making math meaningless.

Arithmetic is the art of processing numbers.
We have ADD, SUBTRACT, MULTPLY and DIVIDE
In ordinary language these words are verbs which have a direct object and an indirect object.

“Add the OIL to the EGG YOLKS one drop at a time”.
“To find the net return subtract the COSTS from the GROSS INCOME”.

In math things have got confused.
We can say “add 3 to 4″or we can say “add 3 and 4”.
We can say “multiply 3 by 4” or we can say “multiply 3 and 4”.
At least we don’t have that choice with subtract or divide.

The direct + indirect form actually means something with the words used,
but when I see “add 3 and 4” my little brain says “add to what?”.

There are perfectly good ways of saying “add, or multiply, 3 and 4” which do not force meanings and usages onto words that never asked for them.
“Find the sum of 3 and 4” and “Find the product of 3 and 4” are using the correct mathematical words, which have moved on from “add” and “multiply”, and incorporate the two commutative laws.

If we were to view operations with numbers as actions, so that an operation such as “add” has a number attached to it, eg “add 7”, then meaningful arithmetical statements can be made, like

“start with 3 and then add 5 and then add 8 and then subtract 4 and then add 1”

which with the introduction of the symbols “+” and “-“, used as in the statement above allows the symbolic expression 3+5+8-4+1 to have a completely unambiguous meaning. It uses the “evaluate from left to right” convention of algebra, and does not rely on any notion of “binary operation” or “properties of operations”.

If we want to view “+” as a binary operation, with two inputs then, yes, we can ascribe meaning to “3+4”, but not in horrors such as the following (found in the CCSSM document):

To add 2 + 6 + 4, the second two numbers can be added to make a ten,
so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

If + is a binary operation, which are the two inputs for the first occurrence of + and which are the inputs for the second occurrence of + ?
The combination of symbols 2 + 6 + 4 has NO MEANING in the world of binary operations.

See A. N. Whitehead in “Introduction to Mathematics” 1911.
here are the relevant pages:

And here are two more delights from the CCSSM document
subtract 10 – 8
add 3/10 + 4/100 = 34/100

In addition I would happily replace the term “algebraic thinking” in grades 1-5 by”muddled thinking”.

The rest are not definitions, though very popular (this is just a selection). But they are true statements

Mathwarehouse: The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a
parabola.
Mathopenref: A parabola is the shape defined by a quadratic equation. The vertex is the peak in the curve as shown on
the right. The peak will be pointing either downwards or upwards depending on the sign of the x2 term.
Virtualnerd: Each quadratic equation has either a maximum or minimum, but did you that this point has a special name?
In a quadratic equation, this point is called the vertex!
Mathwords: Vertex of a Parabola: The point at which a parabola makes its sharpest turn.
Purplemath: The point on this axis which is exactly midway between the focus and the directrix is the “vertex”; the vertex is the point where the parabola changes direction.
Wikibooks: One important point on the parabola itself is called the vertex, which is the point which has the smallest distance between both the focus and the directrix. Parabolas are symmetric, and their lines of symmetry pass through the vertex.
Hotmath: The vertex of a parabola is the point where the parabola crosses its axis of symmetry

Scoring is 10 points for finding the garbage definition and 5 points for the correctish definition !!!! Go for it!

When I studied parabolas, back in 1958 or so (!) the parabola had an apex. So I checked the meaning of vertex, and found that the word was frequently misused.

Basically a vertex of a curve is a point where the curvature is a maximum or a minimum (in non math terms, most or least curved).

Here are two fourth degree polynomials, one has three vertices and the other has five. The maximum curvature points are indicated. The minimum curvature points are at the origin for the first curve, and at the points of inflexion for the second curve (curvature = zero)

A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.

For a circle, which has constant curvature, every point is a vertex.

The center of curvature at a (nice) point on a curve is the center of the closest matching circle at that point. This circle will usually lie “outside” the curve on one side of the point, and “inside” the curve on the other side. Look carefully at the picture. It is called the osculating or kissing circle (from the Latin).

The center of curvature can be estimated by taking two point close to the point of interest, finding the tangents at these points, and then the lines at right angles to them and through the points. the center of curvature is roughly at the point of intersection of these two lines

The diagram below shows this estimate, for the blue parabola, at the vertex.

Finally (this has gone on further than expected!) I found this delightful gif.

Dividing top and bottom by cos(φ)sin(θ) and skipping some tedious algebra we get

tan(φ – θ) = (tan(φ) – tan(θ))/(1 + tan(φ)tan(θ))

This is where the books stop, which turns out to be a real shame !

Going back to the two lines and their equations, the two lines

ax + by = 0 and px + qy = 0

have the same angle between them (some things are toooo obvious)

Things are simpler if we look at these two lines through the origin when they both have positive slope.

Take b and q as positive and write the equations as ax – by = 0 and px – qy = 0

Then the point whose coordinates are (b,a) lies on the first and (q,p) lies on the second.

Also, the slopes of the two lines are now a/b , tan(θ) and p/q , tan(φ)

Let us put these into the tan(φ – θ) equation above, and once more after tedious algebra

tan(φ – θ) = (bp – aq)/(ap + bq)

which is a very nice formula for the tan of the angle between two lines.

This is ok if we are interested just in “the angle between the lines”, but if we are considering rotations, and one of the lines is the “first” one, then the tangent is inadequate. We need both the sine and the cosine of the angle to establish size AND direction (clockwise or anticlockwise).

The formula above can be seen as showing cos(φ– θ) as (ap + bq) divided by something

and more tedious algebra and some “observation and making use of structure” gives

M2 = (a2 + b2)(p2 + q2)

and we now have

sin(φ– θ) = (bp – aq)/M and cos(φ– θ) = (bq + ap)/M

and M is the product of the lengths of the two line segments, from the origin to (b,a) and from the origin to (q,p)

It was at this point that I saw M times the sine of the “angle between” as twice the well known formula for the area of a triangle. “half a b sin(C)”, or, if you prefer, the area of the parallelogram defined by the two line segments.

Suddenly I saw all this in 2D vector terms, with bq + ap being the dot product of (b,a) and (q,p) , and bp – aq as being part of the definition of the 3D vector or cross product, in fact the only non zero component (and in the z direction), since in 3D terms our two vectors lie in the xy plane.

Why is the “vector product” not considered in the 2D case ??? It is simpler, and looking at the formula for sine , above, we have a 2D interpretation of the “vector”or cross product as twice the area of the triangle formed by (b,a) and (q,p). (just as in the standard 3D definition, but treated as a scalar).

So “bang goes” the common terms, scalar product for c . d and vector product for c X d

Dot product and cross product are much better anyway, and a bit of ingenuity will lead you to the reason for the word “cross”.

This is one of the things implemented using this approach:

Anyway, the end result of all this, for rotating points on a circle, was a calculation process which did not require the actual calculation of any angle. No arctan( ) !

Subtraction. Read this
————-
Kindergarten
Operations and Algebraic Thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
————-
What has subtraction got to do with taking apart ???
(The examples are all of the form 9 = 3 + 6 and so on).

Also there is NO mention at all of subtraction as a way of finding the difference between two numbers, or of finding the larger of two numbers (anywhere).

While I am in critical mode I offer two more, less awful, horrors from Grade 1:

“To add 2 + 6 + 4,…” and “For example, subtract 10 – 8″.

The poor symbols are clearly in great pain at this point. Just read aloud exactly what is written…..

The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.
Example:
Take 3 x 7. We know that the value of this is 21
Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c
Then 3 x (2 + 5) has the value 21
But so does 3 x 2 + 3 x 5. To check, get out the blocks !
So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !

Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and here it is: